Short answer: the pages are indexed in advance and these indices are stored in a database; the user's specified search terms are compared to the indices; the results are fairly speedy.

Long answer (or excuse for lack thereof): web searching is a hard and complicated problem. Here's a decent introductory article. Do some web searching on the subject. You'll find a lot of stuff. And if you had a good and novel approach to making it more effective or efficient, Google would probably be very happy to pay you handsomely.